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Numerical analysis and linear theory of pulsatile flow in cylindrical deformable tubes: The testing of a numerical model for blood flow calculation

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Abstract

The purpose of this paper is to present standard results on the effect of nonlinearities on the computed pulsatile flow in a cylindrical distensible tube as a first stage in the calculation of flow in a tapered tube and blood flow in arteries. The calculations are made using the pressure-radius relationship of a rubber tube with no longitudinal motion and for a linearised relationship.

The one-dimensional equations of motion are solved by the method of finite differences. The values of skin friction that are incorporated are determined from the vorticity and continuity equations for a rigid tube and a correction made to the current diameter at each time step. The accuracy of the results is assessed and the effect of varying parameters investigated.

The method is applied to a segment of an infinite tube for which the linear analytical solution is available. The characteristics of the velocity wave calculated from an input pressure wave are presented as departures from the linear theory values of these characteristics, the wave speed, flux and transmission factor per wavelength. Computations are made at values of non-dimensional frequency (Stokes number α) of about 3 and 10.

It is concluded that as far as physiological application is concerned (i.e. small amplitude and long wavelength) the results of linear theory are a very good first approximation for the cylindrical tube. At α=10, the relative departure of wave speed is about 0·5 times the relative diameter amplitude (\({{\hat D} \mathord{\left/ {\vphantom {{\hat D} {\bar D}}} \right. \kern-\nulldelimiterspace} {\bar D}}\) amplitude/mean diameter) when the pressure-radius relation is linear and the pressure and velocity waves have the same characteristics. At α=3 the corresponding wave speed departure is about 0·1\({{\hat D} \mathord{\left/ {\vphantom {{\hat D} {\bar D}}} \right. \kern-\nulldelimiterspace} {\bar D}}\). The relative departure of the flux is less than 0·05\({{\hat D} \mathord{\left/ {\vphantom {{\hat D} {\bar D}}} \right. \kern-\nulldelimiterspace} {\bar D}}\) at α=3 and about 0·5\({{\hat D} \mathord{\left/ {\vphantom {{\hat D} {\bar D}}} \right. \kern-\nulldelimiterspace} {\bar D}}\) at α=10. The transmission coefficient has a relative departure of less than 0·05\({{\hat D} \mathord{\left/ {\vphantom {{\hat D} {\bar D}}} \right. \kern-\nulldelimiterspace} {\bar D}}\) at α=10 and its relative increase at α=3 is about 0·3\({{\hat D} \mathord{\left/ {\vphantom {{\hat D} {\bar D}}} \right. \kern-\nulldelimiterspace} {\bar D}}\).

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Abbreviations

a :

inviscid small amplitude wave speed,a 2=D ∂p/2p ∂D

a 0 :

Zero excess pressure value ofa

c 0 :

Moenz-Korteweg wave speed,c 20 =Eh 0/ρD 0

dt :

Time step

dx :

Distance step in axial direction

D 0 :

Zero excess pressure value of inside diameter of tube

\(\bar D\) :

Mean value of inside diameter of tube

\(\hat D\) :

Amplitude of inside diameter of tube

F 10 :

2J 1(αi3/2){αi3/2Jo(αi3/2)} 1-F10=M10exp(iɛ10

h 0 :

Tube wall thickness at zero excess pressure

L :

Length of tube segment

M :

Number of steps along tube radius

p :

Pressure

\(\hat p_0 \) :

Amplitude of pressure atx=0

r :

Radial distance

Re :

Mean-diameter based Reynolds number, V\({{\bar D} \mathord{\left/ {\vphantom {{\bar D} v}} \right. \kern-\nulldelimiterspace} v}\)

t :

Time

T :

Period of oscillation

v :

Axial velocity,v=v(r)

V :

Velocity averaged over cross-section of tube

\(\hat V\) :

Amplitude ofV

w :

Wave speed

x :

Axial distance

α:

Nondimensional frequency or Stokes number,\(\alpha = 0 \cdot 5\bar D\sqrt {{\omega \mathord{\left/ {\vphantom {\omega v}} \right. \kern-\nulldelimiterspace} v}} \)

Δ:

Divergence from linear theory, for the quantityy Δy=(y c−yl)/yl, wherey c=calculated value andy l=linear theory value

η:

Vorticity, η=−∂v/∂r

λ:

Wavelength

exp (−kλ):

Transmission factor per wavelength

v :

Kinematic viscosity

ψ:

Stokes stream function,v=−1 ∂ψ/r ∂r

τ:

Skin friction stress on the fluid at the wall

θ 0 :

Thickness factor,θ 0=(1+h0/D0)2

ω:

Angular frequency

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  1. Department of the Mechanics of Fluids, University of Manchester, M13 9PL, Manchester, England

    J. H. Gerrard

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Gerrard, J.H. Numerical analysis and linear theory of pulsatile flow in cylindrical deformable tubes: The testing of a numerical model for blood flow calculation. Med. Biol. Eng. Comput. 20, 49–57 (1982). https://doi.org/10.1007/BF02441850

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  • DOI: https://doi.org/10.1007/BF02441850

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